|
| |
|
|
A156547
|
|
Decimal expansion of the central angle of a regular dodecahedron.
|
|
1
|
|
|
|
7, 2, 9, 7, 2, 7, 6, 5, 6, 2, 2, 6, 9, 6, 6, 3, 6, 3, 4, 5, 4, 7, 9, 6, 6, 5, 9, 8, 1, 3, 3, 2, 0, 6, 9, 5, 3, 9, 6, 5, 0, 5, 9, 1, 4, 0, 4, 7, 7, 1, 3, 6, 9, 0, 7, 0, 8, 9, 4, 9, 4, 9, 1, 4, 6, 1, 8, 1, 8, 8, 9, 9, 6, 6, 6, 7, 6, 7, 1, 3, 8, 7, 9, 5, 4, 8, 3, 4, 0, 7, 8, 1, 9, 4, 7, 3, 5, 0, 0, 2, 0, 8, 0, 9, 5
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
If A and B are neighboring vertices of a regular dodecahedron having center O, then the central angle AOB is this number; the exact value is arccos((1/3)*sqrt(5)).
The (minimal) central angle of the other four regular polyhedra are as follows:
- tetrahedron: A156546,
- cube: A137914,
- octahedron: A019669,
- icosahedron: A105199.
|
|
|
LINKS
|
Table of n, a(n) for n=1..105.
|
|
|
FORMULA
|
The dodecahedron has 12 faces and 20 vertices. To find the central angle, we need any neighboring pair of vertices. Here are all 20 vertices:
- (d,d,d) where d is 1 or -1 (that's 8 vertices);
- (0, d*(t-1),d*t), where d is 1 or -1 and d = golden ratio = (1+sqrt(5))/2;
- (d*(t-1), d*t, 0); and ((d*t,0,d*(t-1)).
An example of a neighboring pair is (1,1,1) and (0,t,t-1).
Apply the usual formula for the cosine of the angle between two vectors.
|
|
|
EXAMPLE
|
arccos((1/3)*sqrt(5))=0.729727656226966..., or, in degrees,
41.810314895778598065857916730578259531014119535901347753...
|
|
|
CROSSREFS
|
Sequence in context: A216186 A021936 A154176 * A180872 A003673 A021141
Adjacent sequences: A156544 A156545 A156546 * A156548 A156549 A156550
|
|
|
KEYWORD
|
nonn,cons
|
|
|
AUTHOR
|
Clark Kimberling, Feb 09 2009
|
|
|
STATUS
|
approved
|
| |
|
|
